So, i let my kitchen just go wild cuz it's unruly as hell, but when I used to wrap my hair and wanted it straight I would do this:
use a comb to make a horizontal part right above the kitchen to separate it from the rest of your hair (u can include a tiny bit of 'non kitchen" hair too)
Wrap your hair as normal. (keeping the kitchen part OUT)
lightly oil the kitchen and boar bristle brush it smooth
then with your hand smooth it UP towards the top of your head, flat against the wrapped hair part (so it's going up and your wrapped hair is going horizontal right..)
bobby pin the kitchen hair so it stays up.
put your scarf on covering errythang up and go to sleep.
when you wake up, don't forget to unpin the kitchen first, before you unwrap the rest of your hair.
your kitchen will (should) be smooth and not dry.
It worked for me, i hope it works for you.